66 ( '. Bancs — flotation of Interference Fringes 



shews that for normal incidence, i = 0, the displacement per 

 fringe S< would be 



8t =l_c,>s0' 



or the fringes arc similar to the coarse set described elsewhere.* 

 If the rays impinge at an angle i, figure 2, they will be 

 parallel after the two diffractions are completed; for it is 

 obvious that the corresponding angles of incidence and diffrac- 

 tion are merely exchanged at the two gratings. Hence the 

 homogeneous rays /', impinging at an angle i, leave the grat- 

 ing at J)/ and D" in parallel, at an angle of diffraction i, and 

 the rays unite into a bright image of the slit. If however 

 OG' be displaced a distance, e l to O x G" , parallel to itself, as in 

 figure 2, the paths intercepted are 



— r and '— ■ cos (6 — i) 

 cosz cos^ 



and the path difference per fringe therefore 



X cos i 



he = 



1 — cos (0 — i) ' 



which reduces to the preceding equation if i = 0. Hence a 

 series of interference fringes of the color X must appear in the 

 principal focus of the telescope or lens on either side of i = 0. 

 The theory of diffraction again annuls the apparent path dif- 

 ference between GG and G'G". 



As to the number of fringes, n. between any two angles of 

 incidence i and i\ it appears that n vanishes with e, or the 

 fringes become infinitely large. 



If the grating G' is rotated over an angle 0, fig. 2, and 

 e = b<f>, where b is half the virtual distance apart at the grating 

 G' of the rays impinging upon it, the rotation per fringe is 



X cosi 



b ] — cos (0 — i) ' 



Again, n (above) passes through zero as $ or b decreases from 

 positive to negative values. Variable b implies a wedge effect 

 superposed on the interferences. 



It is this passage of n through zero that is accompanied by 

 the rotation of the fringes, as observed. 



In case of two independent gratings GG and G'G" (G'G" to 

 be treated as consisting of two identical halves OG' and G"0), 

 nearly in parallel, fringes may be modified by rotating G'G" 

 around the three cardinal axes passing through the point of 

 symmetry 0. The rotations of G'G" around an axis normal 



* Phys. Eeview, vii, pp. 79-86, 1916. 



